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Math Help - Concept of the Limit

  1. #1
    Senior Member DivideBy0's Avatar
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    Concept of the Limit

    When I first learned about the limit it greatly disturbed my algebraic thinking, but I pushed the thought aside for a while, and now it's come back.

    Take for example:

    \lim_{x \to 2}\frac{x^2-4}{x^2-2x}

    We have the expression

    \frac{x^2-4}{x^2-2x}, which is equivalent to \frac{x+2}{x}

    On the left, substituting in x we get an undefined answer, substituting into the right expression however, we get 2.

    Aren't the two expression as perfect as each other? Then why do they give a different result when you substitute in x?
    It's like we are, by manipulating the number, forcing it to give a certain answers against its will.
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  2. #2
    MHF Contributor red_dog's Avatar
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    When x\to 2 that means that x is in a neighbourhood of 2 and  x\neq 2.
    So the expression \displaystyle\frac{x^2-4}{x^2-2x} can be simplify by x-2, because  x\neq 2.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by DivideBy0 View Post
    When I first learned about the limit it greatly disturbed my algebraic thinking, but I pushed the thought aside for a while, and now it's come back.

    Take for example:

    \lim_{x \to 2}\frac{x^2-4}{x^2-2x}

    We have the expression

    \frac{x^2-4}{x^2-2x}, which is equivalent to \frac{x+2}{x}
    Only when x \ne 2, because to get the second you will have divided the first by zero (which is not allowed). So while this is true when x \ne 2 you cannot just substitute 2 into both sides and expect equality.

    This is one of the reasons you are not allowed to divide by zero.

    RonL
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